Particle limiting velocities from the bicubic equation derived from Einstein’s kinematics: PeV electron case

摘要

Here one reviews the particle limiting velocities derived from the Einstein’s kinematic generated bicubic equation: c1, the primary limiting velocity, c2, the obscure limiting velocity and c3, the normal limiting velocity. While c1 and c3 are real c2 is imaginary. Each of these limiting velocities depend on particle physics parameters of energy, E, mass, m, and ordinary velocity, v in such a way that c21 , c22 and c23 are all related to each other by simple transforms, leaving invariant the zero sum rule, c21 + c22+ c23= 0. As such, they form the bicubic particle kinematics. Now for the problem at hand, the limiting velocities are calculated specifically for the 0.5 MeV mass electron in the PeV energy region from the 2010 Crab Nebula Flare. Of the three solutions, c1; c2 and c3 one finds c1 to be very large and likely unphysical, similarly imaginary c2 with very large absolute value also likely unphysical and both of them Lorentz violating (LV), while the calculated normal limiting velocity c3 has acceptable values in this high energy case. With the electron energy in the PeV region, the electron mass has very little influence on c3. Even so one calculates miniscule subluminal and superluminal Lorentz violations, when respectively c3 ? c and c3 ? c, and the Lorentz invariance (LI) when the evaluation yields c3 = c. Qualitatively, because of miniscule masses, the calculated electron limiting velocity due to the Crab Nebula Flare PeV events shows great deal of similarities with the calculated neutrino limiting velocity from the OPERA neutrino velocity experiments. To get bigger mass effects on limiting velocities, one needs to go from the energy region E ? mc2 to the energy region E ? mc2 . With this, one would see whether in the lower energy region one has also c3 ? O(c) with LI and LV small portions. Also lower energy electron velocities, may even provide physical reasons for the existance or non-existance of c1 and c2.